A and the vanishing topology of discriminants

Summary Suppose thatf: ⁿ , 0 ^p , 0 is finitely -determined withnp. We define a Milnor fiber for the discriminant off; it is the discriminant of a stabilization off. We prove that this discriminant Milnor fiber has the homotopy type of a wedge of spheres of dimensionp–1, whose number we denote byµ (f). One of the main theorems of the paper is a = type result: if (n, p) is in the range of nice dimensions in the sense of Mather, then -codium,with equality iff is weighted homogeneous. Outside the nice dimensions we obtain analogous formulae with correction terms measuring the presence of unstable but topologically stable germs in the stabilization. These results are further extended to nonlinear sections of free divisors.Oblatum 15-VIII-1990Partially supported by a grant from the National Science Foundation and a Fullbright Fellowship     

On the statistical derivation of the Schrödinger equation

The transition from reversible microscopic operator equations to irreversible equations for a deterministic density matrix is considered for examples of simple systems—the hydrogen atom or a free electron in an electromagnetic field. As a result of the transition, a system of a particle and field oscillators is replaced by a continuous medium. The Schrödinger equation for the deterministic wave function also describes the evolution of a continuum but without allowance for dissipative terms. In this sense, there is an analogy between the Schrödinger equation in quantum mechanics and Euler's equation in hydrodynamics. The smallest size of a point of a continuous medium is described by the classical electron radiusr _e . It also determines the effective Thomson cross section for scattering of photons by free electrons. The lengthr _e and the corresponding time interval _e =r _e /c play the role of hidden parameters in quantum mechanics. Two methods of calculating the effective Thomson cross section in terms of the extinction coefficient are considered. The first of them is based on the equation of motion of a free electron in a field with allowance for radiative friction. This equation leads to well-known difficulties. Moreover, the velocity fluctuations calculated on its basis lead to a contradiction with the second law of thermodynamics. The second method is based on the introduction of a constant friction coefficient = _e ^–1 , the presence of which reflects loss of information on smoothing over the volume of a point of the continuous medium. Such a method of calculation leads to the same expression for the effective cross section but makes it possible to avoid the difficulties with the second law of thermodynamics. In the derivation of quantum kinetic equations, the physically infinitesimally small scales are determined by the Compton length _C and the corresponding time interval. The introduction of these scales makes it possible to separate and eliminate small-scale fluctuations, the collision integrals being expressed in terms of the correlation functions of these fluctuations.In memory of Dmitrii Nikolaevich ZubarevMoscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 1, pp. 3–26, October, 1993.     

Descendants constructed from matter field in Landau-Ginzburg theories coupled to topological gravity

It is argued that gravitational descendants in the theory of topological gravity coupled to topological Landau-Ginzburg theory (not necessarily conformal) can be constructed from matter fields alone (without metric fields and ghosts). In this sense topological gravity is induced. We discuss the mechanism of this effect (that turns out to be connected with K. Saito's higher residue pairing: Kⁱ(_i(₁),₂)=K⁰(₁,₂)), and demonstrate how it works in a simplest nontrivial example: correlator on a sphere with four marked points. We also discuss some results on k-point correlators on a sphere. From the idea of induced topological gravity it follows that the theory of pure topological gravity (without topological matter) is equivalent to the trivial Landau-Ginzburg theory (with quadratic superpotential).Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 2, pp. 307–316, May, 1993.     

A Note on a Theorem of Horst Sachs

In this note, we present an apparently new and rather short proof of a celebrated theorem of Horst Sachs characterizing bipartite finite graphs in term of their eigenvalue spectrum. Moreover, the simplicity of the proof allows us to establish this theorem and related results as a special instance of much more general assertions regarding the spectral theory of compact graphs. Finally, some intriguing possible generalizations to locally finite, yet not compact graphs suggested by Horst Sachs are discussed in the last section.Received December 8, 2003     

Singularities of the Green function of a random walk on a discrete group

LetX be a countable discrete group and let be an irreducible probability onX. The radius of convergence of the Green function is finite, and independent ofx. Let 0} \right\}$$ " align="middle" border="0"> be the period of . We show that for eachxX the singularities of the analytic functionzG(x; z) on the circle {z:|z|=} are precisely the points e ^2ik/d k=0, ...,d–1. In particular, is the only singularity on the circle in the aperiodic cased=1 (which occurs, for example, when (e)>0). This affirms a conjecture ofLalley [5]. When is symmetric, i.e., (x ^–1)=(x) for allxX, d is either 1 or 2. As another particular case of our result, we see that- is then a singularity ofzG (x; z) if and only ifd=2, in which caseX is bicolored. This answers a question ofde la Harpe, Robertson andValette [2].     

О лагранжевом интерполировании с узлами в корнях многочленов сонина-маркова

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Some imbedding theorems concerning the moduli of Ditzian and Totik

, , 3. . . f ^*, f, , f . , .

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Research supported by Hungarian National Foundation for Scientific Research Grant No. 1801.     

On the uniform (C, 1)-summability of Fourier series and integrability of series with respect to multiplicative orthonormal systems

H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp _n=, , ,n .. . , 117 234 . . -      

A Framework for Evaluating Knowledge-Based Interestingness of Association Rules

In Knowledge Discovery in Databases (KDD)/Data Mining literature, interestingness measures are used to rank rules according to the interest a particular rule is expected to evoke. In this paper, we introduce an aspect of subjective interestingness called item-relatedness. Relatedness is a consequence of relationships that exist between items in a domain. Association rules containing unrelated or weakly related items are interesting since the co-occurrence of such items is unexpected. Item-Relatedness helps in ranking association rules on the basis of one kind of subjective unexpectedness. We identify three types of item-relatedness – captured in the structure of a fuzzy taxonomy (an extension of the classical concept hierarchy tree). An item-relatedness measure for describing relatedness between two items is developed by combining these three types. Efficacy of this measure is illustrated with the help of a sample taxonomy. We discuss three mechanisms for extending this measure from a two-item set to an association rule consisting of a set of more than two items. These mechanisms utilize the relatedness of item-pairs and other aspects of an association rule, namely its structure, distribution of items and item-pairs. We compare our approach with another method from recent literature.     

On the divergence of certain Hermite-Fejér interpolation

Summary In his paper [1]P. Turán discovers the interesting behaviour of Hermite-Fejér interpolation (based on the ebyev roots) not describing the derivative values at exceptional nodes {_n} _n=1 . Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever –1<x<1 but does not converge for a suitable continuous function at any point of the whole interval [–1, 1].     

Regularity,partial regularity,partial information process,for a filtered statistical model

Summary We define partial regularity for a filtered statistical (semi-parametric) model indexed by ^d , as differentiability in a suitable sense of the partial likelihoods associated with a basic processX. Partial regularity turns out to be equivalent to some sort of differentiability in of the characteristics ofX. We also prove that regularity of the model implies partial regularity, and we define a partial information process, which is smaller than the complete information process. We apply these results to obtain a generalization of Cramer-Rao inequality, and to prove that partial likelihood processes are optimal among all quasi-likelihood processes which are stochastic integrals with respect to the basic processX.     

Жизнь и деятельность яноша Бояи

: . BOLYAI.     

Local extrema of analytic functions

We give a complete answer to the problem of the finite decidability of the local extremality character of a real analytic function at a given point, a problem that found partial answers in some works by Severi and ojasiewicz. Consider a real analytic functionf defined in a neighbourhood of a pointx ₀R ⁿ . Restrictf to the spherical surface centered inx ₀ and with radiusr0 and take its infimumm(r) and its supremumM(r). We establish some properties ofm(r) andM(r) for smallr>0. In particular, we prove that they have asymptotic expansions of the formf(x ₀)+c·(r +o(r )) asr0 for a realc and a rational 1 (of course the parameters will usually be different form andM).This work was supported by the Brazilian Fundação Carlos Chagas and by the Italian Consiglio Nazionale delle Ricerche.     

On convex multipliers of convergence of some classes of bivariate fourier series

- ()N²,L _F ( ) — , 2- , {s _m() f} -L. — . (L _F( ),L _F( ) ={(k)} (kZ²) , fL_F( ) f , , L _F( ). - ={()} ={()} , n(())m()n(()+()) . R() , .. - . , . (L _F ( ),L _F ( )) , R(,)=O(1) (x).

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The author wishes to express his gratitude to S. A.Teljakovski for setting the problem and for his attention to this paper.     

Absolut stetige Maße auf lokalkompakten Halbgruppen

LetS be a locally compact semigroup. It is shown that if a measure is absolutely continuous and ifS is cancellative, then the measure concentrated on a Borel subsetB ofS (i. e. =(B.)) is also absolutely continuous. Other properties of absolutely continuous measures will be obtained. Moreover we will answer the question when absolutely continuous probability measures exist. This is the case ifS admits an invariant integral on the space of all continuous functions onS with compact support. Another result is the following: If the compact semigroupS has a connected kernel then there exist absolutely continuous probability measures if and only ifS is amenable.     

A result from diophantine approximation which has applications in economics

We say that a real number allows poor approximations if we can find 0<=()<1 and a sequence of integers n₁2<... such that for all rationals p/q with qn. we have |–.p/q| > Kn _j ^–l– where K is a constant depending only on .In this note we prove that the set of numbers which allow poor approximations are precisely the very well-approximable numbers.The existence of numbers with poor approximations has been used by Cheng [1] to show the existence of a dense set of economies whose cone converges to the Walras equilibrium as slowly as 0(n^–1/2–) after n replications.     

Chapter 1 Introduction: Extensions and new developments in DEA

The extensions, new developments and new interpretations for DEA covered in this paper include: (1) new measures of efficiency, (2) new models and (3) new ways of implementing established models with new results and interpretations presented that include treatments of congestion, returns-to-scale and mix and technical inefficiencies and measures of efficiency that can be used to reflect all pertinent properties. Previously used models, such as those used to identify allocative inefficiencies, are extended by means of assurance region approaches which are less demanding in their information requirements and underlying assumptions. New opportunities for research are identified in each section of this chapter. Sources of further developments and possible sources for further help are also suggested with references supplied to other papers that appear in this volume and which are summarily described in this introductory chapter.     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

On the continuity of midpoint hull convex set-valued functions

Summary The concept of hull convexity (midpoint hull convexity) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of convexity (midpoint convexity).The main result is a sufficient condition for a midpoint hull convex set-valued function to be continuous. This theorem improves a result obtained by K. Nikodem (Bulletin of the Polish Academy of Sciences, Mathematics,34 (1986), 393–399).     

On a retract of an integral domain

Let A R be commutative integral domains with identity. We say A is a retract of R when there exists a ring homomorphism : RA such that |_A=id_A. We Prove two theorems: Theorem 3.1 says when A is a retract of R via , with Kernel ()= =t·R (O), then t is transcendental over A and if A is sub-inert in R, then RK(t)=A[t], where K is the quotient field of A. This theorem with the hypotheses strengthened to A being a 2-valuation algebra of R and the transcendence degree of R over A equal to one, gives Theorem 4.1, which now says R=A[t]. Given the other hypotheses of Theorem 4.1, A being a 2-valuation algebra of R is equivalent to the existence of a valuation ring V of L, the quotient field of R, such that VR=A. Some counterexamples are given and some variations on the hypotheses of Theorem 4.1 are discussed.